Palo Verde Workshop March 2014. p.1

Define:

Square:

Triangle:

Angle:

Square:

Triangle:

Angle:



From (6/3/2011):

Discuss.

Multiplicative Thinking I - The Broomsticks

You have three broomsticks:

The RED broomstick is three feet long

The YELLOW broomstick is four feet long

The GREEN broomstick is six feet long

  1. How much longer is the GREEN broomstick than the RED broomstick?
  1. How much longer is the YELLOW broomstick than the RED broomstick?
  1. The GREEN broomstick is ______times as long as the YELLOW broomstick.
  1. The YELLOW broomstick is ______times as long as the GREEN broomstick.
  1. The YELLOW broomstick is ______times as long as the RED broomstick.
  1. The RED broomstick is ______times as long as the YELLOW broomstick.
  1. A certain stock started at a value $74. One year later it was valued at $89.54. By what percent did the stock’s value increase?
  1. The Willis tower (formerly the Sears tower) is 1730 feet high. The Burj Khalifa (formerly Burj Dubai) is 2717 feet high.
  1. The Burj is ______times as large as the Willis tower.
  1. The Willis tower is ______times as large as the Burj
  1. The Burj is ______percent the size of the Willis tower.
  1. The Willis tower is ______percent the size of the Burj.
  1. How can you think about ?
  1. If you are given , how can you think about c?
  1. On the broomstick questions (#3-#6) we did not need to mention “feet” in the answer. Why is that?









Geometric Fractions: Area, Length, Sets.

Use pattern blocks to answer questions 1-9.

  1. If = 1 what is ?
  1. If =1 what is?
  1. If = ¼ what is ?
  1. If = ¼ what is ?
  1. If =2 what is ?
  1. If = 2/3 what is 1? Explain
  1. If = 5/2 what is 1? What is 3/4? Explain.
  1. If = 7 what is ?
  1. If= 1 what is 5/3 ?
  1. If = 3/5 what is 1?
  1. Use your thinking in #9 to explain and finish:
  2. Use your thinking in #9 to explain and finish
  3. Use your thinking in #10 to explain and finish
  1. If segment AB has a length of 7/4, how could you determine a length of 1?
  1. Use your thinking in #14 to explain and finish
  1. Provide two different ways that you could determine a length of 7/8.
  1. If segment CD has a length of 4/7 how could you determine a length of 1?
  1. Provide two different ways that you could determine a length of 8/7.
  1. The following set is 3/5 of some original set. Can the original set be represented as a rectangular array? Explain without using algebra.
  1. What is 5/3 of the given set?Explain without using algebra.
  1. You are told that the following set is 3/5 of some original set. Can the original set be represented as a rectangular array? Explain without using algebra.





Similarity

  1. When we say two figures are similar we mean…
  1. You are given the similar triangles below. Using only our meaning of similarity (and no formulas) explain how to find x.
  1. A’B’ is ______times as long as 1/7 of AB.
  1. So A’B’ is ______times as long as AB.
  1. Since the figures are similar, this means that A’C’ is ______times as long as AC. So A’C’ is ______or ______
  1. How long is is B’C’?
  1. Without using formulas, find x and y in the similar figures:
  1. Bonus! Find the areas of the two figures.

9. You are given the three similar, right triangles below:

  1. Find the length of the hypotenuse for each of the triangles.
  2. Complete each of the following sentences:
  3. In triangle ABC the length of the side opposite angle C is ______times as large as the hypotenuse. That is, it has a length ______“hypotenuse lengths” long.
  4. In triangle A’B’C’ the length of the side opposite angle C’ is ______times as large as the hypotenuse. That is, it has a length ______“hypotenuse lengths” long.
  5. In triangle A’’B’’C’’ the length of the side opposite angle C” is ______times as large as the hypotenuse. That is, it has a length ______“hypotenuse lengths” long.
  6. Will this constant relationship be true for all triangles similar to ABC? Explain.
  1. What is true about the angles at C, C’, and C’’?
  1. If I were to increase the angle at C, C’, and C’’ while keeping the triangle a right triangle, what would happen to the opposite side in terms of hypotenuse-lengths?
  2. If the only length-units we use are “hypotenuse-lengths”, what is the absolute longest the opposite side could ever measure in a right triangle?
  1. What is the absolute shortest?
  1. In a right triangle, this relationship (the opposite side as “how many times as large” as the hypotenuse) is known as “Sine.” Each “Sine” value corresponds to a given acute angle in a right triangle. According to the sine chart on the next page, what should be the approximate measure of angle C?
  1. Measure the angle to confirm.
  1. Use the sine chart to find the value of x in each of the following:


How many cubes does it take to construct each of the following shapes?

4. Briefly describe the thinking you used to answer question 2.

5. A rectangular prism with dimensions 2” x 3” x 4” can have three different bases, each with different dimensions.

a. Find the area of each base.

b. Use each base to find the volume of the rectangular prism.

6. Write a rule to find the volume of any prism using the area of the base and the height of the prism.

7. Explain how the above answers can be applied to determine the volume of any cylinder.

Created by Donna Guhse, Scottsdale Community College,

An admittedly imperfect, yet useful, albeit totally unofficial, graphic organizer for PARCC publications.

Interactive version available at

2013 Ted Coe, Grand Canyon University,